Theorem recexprlemm 7807

Description: B is inhabited. Lemma for recexpr 7821. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemm	|- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemm

Step	Hyp	Ref	Expression
1		prop 7658	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7661	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
3		recclnq 7575	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) e. Q. )
4		nsmallnqq 7595	. . . . . . 7 |- ( ( *Q ` x ) e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
5	3, 4	syl 14	. . . . . 6 |- ( x e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
6	5	adantr 276	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q <Q ( *Q ` x ) )
7		recrecnq 7577	. . . . . . . . . . . 12 |- ( x e. Q. -> ( *Q ` ( *Q ` x ) ) = x )
8	7	eleq1d 2298	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) <-> x e. ( 2nd ` A ) ) )
9	8	anbi2d 464	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) ) )
10		breq2 4086	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( q <Q y <-> q <Q ( *Q ` x ) ) )
11		fveq2 5626	. . . . . . . . . . . . . 14 |- ( y = ( *Q ` x ) -> ( *Q ` y ) = ( *Q ` ( *Q ` x ) ) )
12	11	eleq1d 2298	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) )
13	10, 12	anbi12d 473	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) ) )
14	13	spcegv 2891	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
16	9, 15	sylbird 170	. . . . . . . . 9 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
17		recexpr.1	. . . . . . . . . 10 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
18	17	recexprlemell 7805	. . . . . . . . 9 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
19	16, 18	imbitrrdi 162	. . . . . . . 8 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
20	19	expcomd 1484	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 2nd ` A ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) ) )
21	20	imp 124	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) )
22	21	reximdv 2631	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( E. q e. Q. q <Q ( *Q ` x ) -> E. q e. Q. q e. ( 1st ` B ) ) )
23	6, 22	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q e. ( 1st ` B ) )
24	23	rexlimiva 2643	. . 3 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. q e. Q. q e. ( 1st ` B ) )
25	1, 2, 24	3syl 17	. 2 |- ( A e. P. -> E. q e. Q. q e. ( 1st ` B ) )
26		prml 7660	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
27		1nq 7549	. . . . . . . 8 |- 1Q e. Q.
28		addclnq 7558	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
29	3, 27, 28	sylancl 413	. . . . . . 7 |- ( x e. Q. -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
30		ltaddnq 7590	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
31	3, 27, 30	sylancl 413	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
32		breq2 4086	. . . . . . . 8 |- ( r = ( ( *Q ` x ) +Q 1Q ) -> ( ( *Q ` x ) <Q r <-> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) )
33	32	rspcev 2907	. . . . . . 7 |- ( ( ( ( *Q ` x ) +Q 1Q ) e. Q. /\ ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
34	29, 31, 33	syl2anc 411	. . . . . 6 |- ( x e. Q. -> E. r e. Q. ( *Q ` x ) <Q r )
35	34	adantr 276	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
36	7	eleq1d 2298	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) <-> x e. ( 1st ` A ) ) )
37	36	anbi2d 464	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) ) )
38		breq1 4085	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( y <Q r <-> ( *Q ` x ) <Q r ) )
39	11	eleq1d 2298	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) )
40	38, 39	anbi12d 473	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) ) )
41	40	spcegv 2891	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
42	3, 41	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
43	37, 42	sylbird 170	. . . . . . . . 9 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
44	17	recexprlemelu 7806	. . . . . . . . 9 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
45	43, 44	imbitrrdi 162	. . . . . . . 8 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
46	45	expcomd 1484	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 1st ` A ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) ) )
47	46	imp 124	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) )
48	47	reximdv 2631	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( E. r e. Q. ( *Q ` x ) <Q r -> E. r e. Q. r e. ( 2nd ` B ) ) )
49	35, 48	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. r e. ( 2nd ` B ) )
50	49	rexlimiva 2643	. . 3 |- ( E. x e. Q. x e. ( 1st ` A ) -> E. r e. Q. r e. ( 2nd ` B ) )
51	1, 26, 50	3syl 17	. 2 |- ( A e. P. -> E. r e. Q. r e. ( 2nd ` B ) )
52	25, 51	jca 306	1 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   <.cop 3669   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   1stc1st 6282   2ndc2nd 6283   Q.cnq 7463   1Qc1q 7464   +Q cplq 7465   *Qcrq 7467   <Q cltq 7468   P.cnp 7474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-inp 7649

This theorem is referenced by:  recexprlempr  7815